Page translated by Claude — switch to Italian to read the original article.

Anno Domini 2020, December: media all over the world take an interest in a rare astronomical event — the conjunction between Jupiter and Saturn.

Conjunctions between these two planets recur at intervals of about 20 years, but this one had a special feature: at the moment of closest approach the two planets would be just over 6 arcminutes apart.

A separation of 6’ means that the two planets can be observed and photographed at the same time within the telescope’s field of view.

The interesting thing is that this makes it easy to compare the two planets directly: their apparent size and their brightness.

It was precisely this feature that inspired this simple exercise.

Unfortunately the weather in my area was unkind and I was unable to observe the event, but social media were literally flooded with more or less beautiful and interesting images of it.

Among the many, I was struck by an image by Alessandro Bianconi, without doubt one of the best planetary astrophotographers in the world,

In his image the two planets appear close together and you can see a clear difference in the brightness of the planetary surfaces lit by the Sun.

The image can be seen on Alessandro’s Astrobin profile (https://astrob.in/ssqwqp/0/).

As everyone knows, planets do not emit light (in the visible spectrum) but reflect the light received directly from the Sun.

The intensity of sunlight, to a first approximation, decreases as the inverse square of the distance, so if the distance doubles, the light received from the Sun is reduced to a quarter.

This difference is clearly visible in Alessandro’s image, as can be seen from the detail extracted from his photo.

The surface luminosity of an object is called surface brightness.

A remarkable property of surface brightness is that it does not depend on the distance of the observer: a distant observer will see the object as smaller than a nearby one, but the surface brightness will be the same.

In practice, therefore, the surface brightness of a planet depends on just two parameters:

• the distance from the Sun
• the albedo of the surface (that is, its ability to reflect light).

At equal distance, a grey object will obviously appear darker than a completely white one.

From these few considerations comes the idea for the experiment: assuming that Jupiter and Saturn have the same albedo and that the image sensor I use to capture them is linear, it is possible, by measuring the surface brightness, to estimate the ratio between their distances from the Sun.

With a few simple calculations it is easy to show that

where the D’s indicate the distances and the B’s the surface brightnesses.

Clearly the initial hypothesis that the two planets have the same albedo is, at best, a rough approximation; however, since they are both gas giants of comparable size, I expect the difference to be limited.

This is certainly not a true measurement, but rather an estimate.

And here is the procedure used for the experiment:

I opened Alessandro’s original, unprocessed image in PixInsight

Using the Statistics process I extracted:

• Mean and standard deviation of a small region at the centre of Jupiter's disc
• Mean and standard deviation of a small region at the centre of Saturn's disc
• Mean and standard deviation of a region of background sky, in order to determine the background signal to be subtracted from the brightnesses so as to obtain the net value.

The brightnesses used in the formula above are, of course, the “net” ones, i.e. those from which the background-sky value has been subtracted.

Finally, using Cartes du Ciel, I checked the actual distances of the planets from the Sun and calculated the expected ratio so as to compare it with the estimate made.

The result of the calculations was as follows

Well, our estimate tells us that Saturn is 1.835 times farther from the Sun than Jupiter, with a standard deviation of 0.021

This means we have a 99.7% probability (3-sigma uncertainty) that the value falls within the range 1.771 – 1.900

The actual expected value is 1.959 (with a measurement error of about 6%).

What do these data tell us?

Our estimate is certainly different from the expected value; indeed, the latter falls outside the uncertainty range of the measurement.

This means that one of our initial hypotheses is wrong.

All the clues point precisely to the hypothesis of equal albedo: the difference in the reflectivity of the clouds of the two gas giants introduced an unaccounted-for source of uncertainty.

These data tell us, in effect, that the surfaces of Jupiter and Saturn do not have the same albedo; in fact, our experiment allows us to estimate that difference:

To make the ratio of distances equal to the expected one, it is enough to increase the measured brightness of Jupiter by about 13% so as to “match” that of Saturn.

In other words, this means that, from our measurements, Saturn’s surface must be 13% more reflective than Jupiter’s.

So here we see that, even with a simple photo of one of the most over-hyped astronomical events of 2020, it is possible to do a little bit of “science from home”.

Thanks again to Alessandro Bianconi for granting me his beautiful photo.